How could I have an object's Z axis align with the first derivative of is location F-Curve
If the goal is for an object to move along a curve (e.g. Bezier) with one of its axis aligned with the curve's first derivative, as suggested by @atomicbezierslinger, you can use a follow path constraint. For example, if we want the object's Z axis to be aligned with the curve's tangent:


Set Follow  Curve to true.
Set the Forward axis to Z
Set the Up axis to either X or Y (setting to Z will not work, and the object might not move at all).